Basics of Quadrilaterals — CAT Previous-Year Questions with Solutions

CAT 2024 Slot 2 · QA

Q1.

ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC when extended, intersect at point E.

If AB = 2 cm, CD = 1 cm, and perimeter of ABCD is 6 cm, then the perimeter, in cm, of âAEB is

CAT 2023 Slot 1 · QA

Q2.

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

CAT 2023 Slot 2 · QA

Q3.

In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is:

CAT 2023 Slot 3 · QA

Q4.

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is?

CAT 2022 Slot 1 · QA

Q5.

A trapezium ABCD has side AD parallel to BC. ∠BAD = 90°, BC = 3 cm and AD = 8 cm. If the perimeter of this trapezium is 36 cm, then its area, in sq. cm, is

CAT 2022 Slot 3 · QA

Q6.

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

CAT 2021 Slot 2 · QA

Q7.

The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If the difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is

CAT 2021 Slot 2 · QA

Q8.

If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is

CAT 2021 Slot 3 · QA

Q9.

A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is

CAT 2021 Slot 3 · QA

Q10.

Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30° then the area of the parallelogram is sq. cm. is

CAT 2020 Slot 1 · QA

Q11.

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

CAT 2020 Slot 2 · QA

Q12.

The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy the relationship R = T². If the sides of the rectangle are in the ratio 1 : 3, then the length, in cm, of the longer side of the rectangle, is

CAT 2020 Slot 3 · QA

Q13.

In a trapezium ABCD, AB is parallel to DC, BC is perpendicular to DC and ∠BAD = 45°. If DC = 5 cm, BC = 4 cm, the area of the trapezium in sq.cm is

CAT 2018 Slot 1 · QA

Q14.

Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is

CAT 2018 Slot 1 · QA

Q15.

In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is

CAT 2018 Slot 1 · QA

Q16.

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

CAT 2018 Slot 2 · QA

Q17.

A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of the following is necessarily true?

CAT 2018 Slot 2 · QA

Q18.

The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio ?

CAT 2017 Slot 2 · QA

Q19.

ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is

CAT 2008 · QA

Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Q20.

Consider a square ABCD with midpoints E, F, G, H of AB, BC, CD and DA respectively. Let L denote the line passing through F and H. Consider points P and Q, on L and inside ABCD, such that the angles APD and BQC both equal 120°. What is the ratio of the area of ABQCDP to the remaining area inside ABCD?

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let a₁ = p and b₁ = q, where p and q are positive quantities.

Define:
a_n = pb_n−1 b_n = qb_n−1, for even n > 1 and
a_n = pa_{n − 1} b_n = qa_{n − 1}, for odd n > 1.

Q21.

Each question is followed by two statements A and B. Answer each question using the following instructions.
Mark (1) if the question can be answered by using statement A alone but not by using statement B alone.
Mark (2) if the question can be answered by using statement B alone but not by using statement A alone.
Mark (3) if the question can be answered by using both the statements together but not by using either of the statements alone.
Mark (4) if the question cannot be answered on the basis of the two statements.

Rahim plans to draw a square JKLM with a point O on the side JK but is not successful. Why is Rahim unable to draw the square?

A. The length of OM is twice that of OL.
B. The length of OM is 4 cm.

CAT 2006 · QA

Passage / Data

Answer the following question based on the information given below.

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs. 1200 and Rs. 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs. 5400.

Q22.

An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?

CAT 2003 Slot 2 · QA

Q23.

A square tin sheet of side 12 inches is converted into a box with open top in the following steps – the sheet is placed horizontally. Then, equal sized squares, each of side x inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If x is an integer, then what value of x maximizes the volume of the box?

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q24.

Let S₁be a square of side a. Another square S₂ is formed by joining the mid-points of the sides of S₁. The same process is applied to S₂ to form yet another square S₃, and so on. If A₁, A₂, A₃... are the areas and P₁, P₂, P₃... are the perimeters of S₁, S₂, S₃... respectively,

then the ratio $\frac{P_{1} + P_{2} + P_{3} + . . . . .}{A_{1} + A_{2} + A_{3} + . . .}$ equals:

CAT 2002 · QA

Q25.

In the following figure, the area of the isosceles right triangle ABE is 7 sq.cm. If EC = 3BE, then the area of rectangle ABCD (in sq. cm.) is

CAT 2001 · QA

Q26.

In the above diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the area of the triangle CEF and that of the rectangle?

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r₁ = number of runs scored in completed innings; n₁ = number of completed innings

r₂ = number of runs scored in incomplete innings; n₂ = number of incomplete innings

BA = $\frac{r_{1} + r_{2}}{n_{1}}$

To better assess batsman's accomplishments, the ICC is considering two other measures MBA₁ and MBA₂ defined as follows:

MBA₁ = $\frac{r_{1}}{n_{1}} + \frac{n_{2}}{n_{1}} \times m a x [0, (\frac{r_{2}}{n_{2}} - \frac{r_{1}}{n_{1}}])$

MBA₂ = $\frac{r_{1} + r_{2}}{n_{1} + n_{2}}$

Q27.

Based on the figure below, what is the value of x, if y = 10?

CAT 2000 · QA

Passage / Data

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q28.

ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

CAT 1999 · QA

Q29.

The figure below shows two concentric circles with centre O. PQRS is a square inscribed in the outer circle. It also circumscribes the inner circle, touching it at points B, C, D and A. What is the ratio of the perimeter of the outer circle to that of polygon ABCD?

CAT 1999 · QA

Passage / Data

Directions : Answer the questions based on the following information.

A rectangle PRSU, is divided into two smaller rectangles PQTU, and QRST by the line TQ. PQ = 10 cm. QR = 5 cm and RS = 10 cm. Points A, B, F are within rectangle PQTU, and points C, D, E are within the rectangle QRST. The closest pair of points among the pairs (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) are $10 \sqrt{3}$ cm apart.

Q30.

Which of the following statements is necessarily true?

CAT 1999 · QA

Passage / Data

Directions : Answer the questions based on the following information.

A rectangle PRSU, is divided into two smaller rectangles PQTU, and QRST by the line TQ. PQ = 10 cm. QR = 5 cm and RS = 10 cm. Points A, B, F are within rectangle PQTU, and points C, D, E are within the rectangle QRST. The closest pair of points among the pairs (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) are $10 \sqrt{3}$ cm apart.

Q31.

AB > AF > BF ; CD > DE >CE ; and BF = 6 $\sqrt{5}$ cm. Which is the closest pair of points among all the six given points?

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A cow is tethered at point A by a rope. Neither the rope nor the cow is allowed to enter ΔABC.

∠BAC = 30°
I(AB) = I(AC) = 10 m

Q32.

Four identical coins are placed in a square. For each coin the ratio of area to circumference is same as the ratio of circumference to area. Then find the area of the square that is not covered by the coins.

CAT 1997 · QA

Passage / Data

Direction: Answer the questions based on the following information.

For these questions the following functions have been defined.

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Q33.

The adjoining figure shows a set of concentric squares. If the diagonal of the innermost square is 2 units, and if the distance between the corresponding corners of any two successive squares is 1 unit, find the difference between the areas of the eighth and the seventh squares, counting from the innermost square.

CAT 1997 · QA

Passage / Data

Direction: Answer the questions based on the following information.

For these questions the following functions have been defined.

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Q34.

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?

CAT 1997 · QA

Passage / Data

Direction: Answer the questions based on the following information.

Boston is 4 hr ahead of Frankfurt and 2 hr behind India. X leaves Frankfurt at 6 p.m. on Friday and reaches Boston the next day. After waiting there for 2 hr, he leaves exactly at noon and reaches India at 1 a.m. On his return journey, he takes the same route as before, but halts at Boston for 1hr less than his previous halt there. He then proceeds to Frankfurt.

Q35.

In the given figure, EADF is a rectangle and ABC is a triangle whose vertices lie on the sides of EADF and AE = 22, BE = 6, CF = 16 and BF = 2. Find the length of the line joining the mid-points of the sides AB and BC.

CAT 1996 · QA

Passage / Data

Direction: Answer the questions based on the following information.

In a locality, there are five small cities: A, B, C, D and E. The distances of these cities from each other are as follows.

AB = 2 km AC = 2km AD > 2 km AE > 3 km BC = 2 km
BD = 4 km BE = 3 km CD = 2 km CE = 3 km DE > 3 km

Q36.

If ABCD is a square and BCE is an equilateral triangle, what is the measure of ∠DEC?

CAT 1996 · QA

Passage / Data

Direction: Answer the questions based on the following information.

A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. But, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs. 1,148, but the inventory reduced by 54.

Q37.

The figure shows the rectangle ABCD with a semicircle and a circle inscribed inside in it as shown. What is the ratio of the area of the circle to that of the semicircle?

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q38.

PQRS is a square. SR is a tangent (at point S) to the circle with centre O and TR = OS. Then the ratio of area of the circle to the area of the square is

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q39.

In the adjoining figure, AC+ AB = 5AD and AC – AD = 8. Then the area of the rectangle ABCD is

CAT 1994 · QA

Passage / Data

Answer the next 2 questions based on the following information:

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Q40.

Four friends start from four towns, which are at the four corners of an imaginary rectangle. They meet at a point which falls inside the rectangle, after travelling distances of 40, 50 and 60 metres. The maximum distance that the fourth could have traveled is (approximately) ….

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Alphonso, on his death bed, keeps half his property for his wife and divide the rest equally among his three sons Ben, Carl and Dave. Some years later Ben dies leaving half his property to his widow and half to his brothers Carl and Dave together, shared equally. When Carl makes his will he keeps half his property for his widow and the rest he bequeaths to his younger brother Dave. When Dave dies some years later, he keeps half his property for his widow and the remaining for his mother. The mother now has Rs. 1,575,000.

Q41.

Data is provided followed by two statements – I and II – both resulting in a value, say I and II.

As your answer,
Type 1, if I > II.
Type 2, if I < II.
Type 3, if I = II.
Type 4, if nothing can be said.

In ΔACD, AD = AC and ∠C = 2∠E. The distance between parallel lines AB and CD is h. Then
I. Area of parallelogram ABCD
II. Area of ΔADE

CAT 1993 · QA

Passage / Data

Answer the following questions based on the information given below:

ABC forms an equilateral triangle in which B is 2 km from A. A person starts walking from B in a direction parallel to AC and stops when he reaches a point D directly east of C. He, then, reverses direction and walks till he reaches a point E directly south of C.

Q42.

Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points?

CAT 1991 · QA

Passage / Data

Use the following information:

Prakash has to decide whether or not to test a batch of 1000 widgets before sending them to the buyer. In case he decides to test, he has two options: (a) Use test I ; (b) Use test II. Test I cost Rs. 2 per widget. However, the test is not perfect. It sends 20% of the bad ones to the buyer as good. Test II costs Rs. 3 per widget. It brings out all the bad ones. A defective widget identified before sending can be corrected at a cost of Rs. 25 per widget. All defective widgets are identified at the buyer’s end and penalty of Rs. 50 per defective widget has to be paid by Prakash.

Q43.

Let the consecutive vertices of a square S be A, B, C & D. Let E, F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to

Basics of Quadrilaterals — CAT Previous-Year Questions

Basics of Quadrilaterals · CAT PYQs